## Beautiful Books

This is an (ever-growing and ever-changing) list of books, useful for school and college mathematics students. If you are working toward Math Olympiad, I.S.I., C.M.I. entrance programs or intense college mathematics, these books may prove to be your best friend.

If you are taking a Cheenta Advanced Math Program, chances are that you will referred to use this post.

## I.S.I. Entrance Interview Problems

1. a and b are two numbers having the same no. of digits and same sum of digits (=28). Can one be a multiple of the other? a is not equal to b. (courtesy Abhra Abir Kundu)
2. Is $$e^x-sinx$$ a polynomial ? (courtesy Tias Kundu)
3. Find the number of onto function from set A containing n elements to set B containing m elements (m<n) (courtesy Tias Kundu)
4. If a+b+c=30,  how many (a,b,c) tuples possible (a,b,c all non-negative). (courtesy Saikat Palit)
5. Can sin(x) be expressed as a polynomial in x? (courtesy Soumik Bhattacharyya)
6. Integers 1-64 are placed in a 8X8 chessboard. How many ways are there to place them such that all numbers in the 1st row and column are in AP? (courtesy Soumik Bhattacharyya)

## Math and Basic Science in India after school

We observe a renewed interest in basic science and mathematics in India after decades of engineering mania. This is a welcome change. The top tier students, in increasing numbers, are opting for such courses.

In this article we present some of the institutes which offer world class education in basic science after school.

## Initiating a child into the world of Mathematical Science

“How do I involve my son in challenging mathematics? He gets good marks in school tests but I think he is smarter than school curriculum.”

“My daughter is in 4th grade. What competitions in mathematics and science can she participate? How do I help her to perform well in those competitions?”

“I have a 6 years old kid. He hates math. How do I change that?”

We often get queries and requests like these from parents around the world. Literally. In fact the first one came from Oregon, United States, second one from Cochin, India and last one from Singapore.

# What is RMO

RMO or Regional Math Olympiad is the first round of mathematics contest (in India) leading to the prestigious International Mathematics Olympiad. It is held in the month of December (first Sunday of December).

The test is conducted in each of the 19 regions of India. From each region about 30 students are selected for the next level (Indian National Math Olympiad). This next level (INMO) is held in February. About 600 students appear in this INMO. Among these 600 students 35 are selected for a one month long camp (International Math Olympiad Training Camp) held in summer in Mumbai.

A flywheel with a radius of $$0.3$$m starts from rest and accelerates with a constant angular acceleration of $$0.6 rads^{-2}$$. Compute the magnitude of the tangential, radial acceleration of a point on its rim at the start.

Solution:

The flywheel has radius $$0.3$$m and starts from rest and accelerates with a  constant angular acceleration of $$0.6 rads^{-2}$$.

The tangential acceleration $$a_{tan}=r\alpha=(0.3)(0.6)=0.18m/s^2$$
Radial acceleration $$a_{rad}=0$$ since the flywheel starts from zero.

## Average Angular Velocity

A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to $$\theta(t)=\gamma t+\beta t^3$$ where $$\gamma=0.4rad/s$$ and $$\beta=0.0120 rad/s^3$$. What is the initial value of the angular velocity? Find the average angular velocity for the time interval $$t=0$$ to $$t=5$$s.
Discussion:

The angle through which the merry-go-round has turned varies with time according to $$\theta(t)=\gamma t+\beta t^3$$ where $$\gamma=0.4rad/s$$ and $$\beta=0.0120 rad/s^3$$.

$$\omega=\frac{d\theta}{dt}$$

At $$t=0$$ $$\omega=\gamma=0.4 rad/s$$
At $$t=0$$, $$\theta=0$$.
At $$t=5$$, $$\theta=(0.4)(5)+(0.012)(5)^3=3.50 rad$$
$$\omega= \frac{3.5-0}{5-0}=0.7 rad/s$$

## Non-existence of continuous function (TIFR 2013 problem 24)

Question:

True/False?

There exists a continuous surjective function from $$S^1$$ onto $$\mathbb{R}$$.

Hint:

Search for topological invariants.

Discussion:

We know that continuous image of a compact set is compact. $$S^1$$ is a subset of $$\mathbb{R}^2$$, and in $$\mathbb{R}^2$$ a set is compact if and only if it is closed and bounded.

By definition, every element of $$S^1$$ has unit modulus, so it is bounded.

Let’s say $$z_n\to z$$ as $$n\to \infty$$. Where {$$z_n$$} is a sequence in $$S^1$$. Since modulus is a continuous function, $$|z_n| \to |z|$$, the sequence {$$|z_n|$$} is simply the constant sequence $$1,1,1,…$$ hence $$|z|=1$$.

What does above discussion mean? Well it means that if $$z$$ is a limit point (or even a point of closure) of $$S^1$$ then $$z\in S^1$$.  Therefore, $$S^1$$ is closed.

The immediate consequence is that the given statement is False. Because, $$\mathbb{R}$$ is not compact. $$S^1$$ is compact, and continuous image of a compact set has to be compact.

## Angular Velocity and Acceleration

A fan blade rotates with angular velocity given by $$\omega=\gamma-\beta t^2$$ where $$\gamma=5$$rad/s and $$\beta=0.800$$rad/s. Calculate the angular acceleration as a function of time.
Solution:

The angular acceleration is given by $$\alpha=\frac{d\omega}{dt}=-2Bt=(-1.60)t$$

The unit of angular acceleration will be $$rad/s^3$$.

## Pre RMO 2017

1. How many positive integers less than $$1000$$ have the property that the sum of the digits of each such number is divisible by $$7$$ and the number itself is divisible by $$3$$ ?
2. Suppose $$a,b$$ are positive real numbers such that $$a\sqrt{a}+b\sqrt{b}=183$$. $$a\sqrt{b}+b\sqrt{a}=182$$. Find $$\frac{9}{5}(a+b)$$.
3. A contractor has two teams of workers: team A and team B. Team A can complete a job in $$12$$ days and team B can do the same in $$36$$ days.Team A starts working on the job and team B joins team A after four days.The team A withdraws after two more days. For how many more days should team B work to complete the job?
4. Let $$a,b$$ be integers such that all the roots of the equation $$(x^2+ax+20)(x^2+17x+b)=0$$ are negative integers.What is the smallest possible value of $$a+b$$?
5. Let $$u,v,w$$ be real numbers in geometric progression such that $$u>v>w$$. Suppose $$u^{40}=v^n=w^{60}$$.Find the value of $$n$$.
6. Let the sum $$\sum_{n=1}^{9}\frac{1}{n(n+1)(n+2)}$$ written in its lowest terms be $$\frac{p}{q}$$.Find the value of $$q-p$$.
7. Find the number of positive integers $$n$$, such that $$\sqrt{n}+\sqrt{n+1}<11$$.
8. A pen costs ₹ $$11$$ and a notebook costs ₹ $$13$$.find the number of ways in which a person can spend exactly ₹ $$1000$$ to buy pens and notebooks.
9. There are five cities $$A,B,C,D,E$$ on a certain island.Each city is connected to every other city by road.In how many ways can a person starting from city $$A$$ come back to $$A$$ after visiting some cities without visiting a city more than once and without taking the same road more than once.? (The order in which he visits the cities also matters; e.g., the routes $${A}\to{B}\to{C}\to{A}$$ and $${A}\to{C}\to{B}\to{A}$$ are different. )
10. There are eight rooms on the first floor of a hotel,with four rooms on each side of the corridor,symmetrically situated (that is each room is exactly oposite to one other room).Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in oposite rooms.In how many ways can the guests be accommodated?
11. Let $$f(x)=sin\frac{x}{3}+cos\frac{3x}{10}$$ for all real $$x$$. Find the least natural number $$n$$ such that $$f(n\pi+x)=f(x)$$ for all real $$x$$.
12. In a class, the total number of boys and girls are in the ratio $$4:3$$. On one day it was found that $$8$$ boys and $$14$$ girls were absent from the class, and that the number of boys was the square of the number of girls. What is the total number of students in the class?
13.  In a rectangle $$ABCD$$. $$E$$ is the midpoint of $$AB$$: $$F$$ is a point on $$AC$$ such that $$BF$$ is perpendicular to $$AC$$: and $$FE$$ perpendicular to $$BD$$. Suppose $$BC=8\sqrt{3}$$. Find $$AB$$.
14. Suppose $$x$$ is a positive real number such that $$\{x\}.[x]$$ and $$x$$ are in a geometric progression. Find the least positive integer $$n$$ such that $$x^n>100$$. (Here $$[x]$$ denotes the integer part of $$x$$ and $$\{x\}=x-[x]$$.)
15. Integers $$1,2,3,…,n$$, where $$n>2$$, are written on a board. Two numbers $$m,k$$ such that $$1<{m}<{n}$$, $$1<{k}<{n}$$ are removed and the averege of the remaining numbers is found to be $$17$$. What is the maximum sum of the two removed numbers?
16. Five distinct $$2-digit$$ numbers are in a geometric progression. Find the middle term.
17. Suppose the altitudes of a triangle are $$10$$, $$12$$ and $$15$$. What is its semi-perimeter?
18. If the real numbers $$x,y,z$$ are such that $$x^2+4y^2+16z^2=48$$ and $$xy+4yz+2zx=24$$, what is the value of $$x^2+y^2+z^2$$?
19. Suppose $$1,2,3$$ are the roots of the equation $$x^4+ax^2+bx=c$$. Find the value of $$c$$.
20.  What is the number of triples $$(a,b,c)$$ of positive integers such that (i) $$a<b<c<10$$ and (ii) $$a,b,c,10$$ form the sides of a quadrilateral?
21. Find the number of ordered triples $$(a,b,c)$$ of positive integers such that $$abc=108$$.
22. Suppose in the plane $$10$$ pairwise nonparallel lines intersect one another. What is the maximum possible number of polygons (with finite areas) that can be formed?
23.  Suppose an integer $$x$$, a natural number $$n$$ and a prime number $$p$$ satisfy the equation $$7x^2-44x+12=p^n$$. Find the largest value of $$p$$.
24. Let $$P$$ be an interior point of a triangle $$ABC$$ whose side lengths are $$26,65,78$$. The line through $$P$$ parallel to $$BC$$ meets $$AB$$ in $$k$$ and $$AC$$ in $$L$$. The line through $$P$$ parallel to $$CA$$ meets $$BC$$ in $$M$$ and $$BA$$ in $$N$$. The line through $$P$$ parallel to $$AB$$ meets $$CA$$ in $$S$$ and $$CB$$ in $$T$$. If $$KL,MN,ST$$ are of equal lengths, find this common length.
25. Let $$ABCD$$ be a rectangle and let $$E$$ and $$F$$ be points on $$CD$$ and $$BC$$ respectively such that area $$(ADE)=16$$ area $$(CEF)=9$$ and area $$(ABF)=25$$. What is the area of triangle $$AEF$$?
26. Let $$AB$$ and $$CD$$ be two parallel chords in a circle with radius $$5$$ such that the centre $$O$$ lies between these chords. Suppose $$AB=6$$, $$CD=8$$. Suppose further that the area of the part of the circle lying between the chords $$AB$$ and $$CD$$ is $$(m\pi+n)/k$$, where $$m,n,k$$ are positive integers with $$gcd(m,n,k)=1$$. What is the value of $$m+n+k$$?
27. Let $${\Omega}_{1}$$ be a circle with centre $$O$$ and let $$AB$$ be a diameter of $${\Omega}_{1}$$. Let $$P$$ be a point on the segment $$OB$$ different from $$O$$. Suppose another circle $${\Omega}_{2}$$ with centre $$P$$ lies in the interior of $${\Omega}_{1}$$. Tangents are drawn from $$A$$ and $$B$$ to the circle $${\Omega}_{2}$$ intersecting $${\Omega}_{1}$$ again at $$A_1$$ and $$B_1$$ respectively such that $$A_1$$ and $$B_1$$ are on the opposite sides of $$AB$$. Given that $${A_1}B=5$$, $$A{B_1}=15$$ and $$OP=10$$, find the radius of $${\Omega}_{1}$$.
28. Let $$p,q$$ be prime numbers such that $$n^{3pq}-n$$ is a multiple of $$3pq$$ for all positive integers $$n$$. Find the least possible value of $$p+q$$.
29. For each positive integer $$n$$, consider the highest common factor $$h_n$$ of the two numbers $$n!+1$$ and $$(n+1)!$$. For $$n<100$$, find the largest value of $$h_n$$.
30.  Consider the areas of the four triangles obtained by drawing the diagonals $$AC$$ and $$BD$$ of a trapezium $$ABCD$$. The product of this areas, taken two at time, are computed. If among the six products so obtained two products are $$1296$$ and $$576$$, determine the square root of the maximum possible area of the trapezium to the nearest integer.

## Angular Velocity

An airplane propeller is rotating at $$1900$$rpm (rev/min).

(a)Compute the propeller’s angular velocity in rad/s.

(b) How many seconds does it take for the propeller to run through $$35^\circ$$?

Solution:

An airplane propeller is rotating at $$1900$$rpm (rev/min).

$$1$$rpm = $$2\pi /60$$$$\omega=(1900)(2\pi /60)=199$$Hence, the propeller’s angular velocity $$\omega$$=$$199$$rad/s.

b) $$35^\circ$$$$\pi/180^\circ$$=$$0.611$$rad.

Since angular velocity $$\omega$$=199rad/s, the time required for the propeller to run through $$35^\circ$$=$$\frac{0.611}{199}=3.1\times10^{-3}s$$

## Complete-Not Compact (TIFR 2013 problem 23)

Question:

True/False?

Let $$X$$ be complete metric space such that distance between any two points is less than 1. Then $$X$$ is compact.

Hint:

What happens if you take discrete space?

Discussion:

Discrete metric space as we know it doesn’t satisfy the distance < 1 condition. But we can make slight changes to serve our purpose.

In $$X$$ define $$d(x,y)=\frac{1}{2}$$ if $$x\ne y$$. Otherwise, $$d(x,x)=0$$.

$$d$$ is indeed a metric, and it gives the same discrete topology on $$X$$. Namely, every set is open because every singleton is open. And therefore every set is closed.

We want $$X$$ to be complete. If $$x_n$$ is a sequence in $$X$$ which is Cauchy, then taking $$\epsilon=\frac{1}{4}$$ in the definition of Cauchy sequence, we conclude that the sequence is eventually constant.

Since the tail of the sequence is constant, the sequence converges (to that constant).

This shows that $$X$$ is indeed Complete.

We don’t want $$X$$ to be compact. Not all $$X$$ will serve that purpose, for example a finite set is always compact. We take a particular $$X=\mathbb{R}$$.

Since singleton sets are open, if we cover $$X$$ by all singleton sets, then that cover has no finite subcover. Hence $$X$$ is not compact.

Therefore the given statement is False.

## Non-homeomorphic (TIFR 2013 problem 22)

Question:

The sets $$[0,1)$$ and $$(0,1)$$ are homeomorphic.

Hint:

Check some topological invariants.

Discussion:

In $$[0,1)$$, $$0$$ seems to be a special point, as compared to $$(0,1)$$ where every point has equal importance (or non-importance).

If $$f:X\to Y$$ is a homeomorphism then for any point $$a\in X$$,  $$X- \{ a \}$$ and $$Y- \{ f(a) \}$$ are homeomorphic with the homeomorphism function being restriction of $$f$$ to $$X- \{a \}$$.

If $$f: [0,1) \to (0,1)$$ is a homeomorphism, then choosing to remove $$0$$ from $$[0,1)$$ we get that $$(0,1)$$ is homeomorphic to $$(0,1)- \{f(0) \}$$.

Whatever be $$f(0)$$, we see that $$(0,1)- \{f(0) \}$$ is a disconnected set. Whereas, $$(0,1)$$ is connected. A continuous image of a connected set is always connected. Hence we are forced to conclude that there was no such $$f$$ to begin with.

So the statement is False.

## A Bouncing Ball

A ball is dropped from a height $$h$$ above a horizontal concrete surface. The coefficient of restitution for the collision involved is $$e$$. What is the time after which the ball stops bouncing?
Discussion:
The time required for the free fall of the ball is $$\sqrt{\frac{2h}{g}}$$. Then the time taken for rise and next fall will be $$2\sqrt{\frac{2h}{g}}e$$.
Time taken for one more rise and fall will be $$2\sqrt{\frac{2h}{g}(e^2)}$$. Thus, the total time for which the ball will be in motion will be
$$\sqrt{\frac{2h}{g}}+2\sqrt{\frac{2h}{g}e(1+e+e^2+….)}$$ $$=\sqrt{\frac{2h}{g}(1+\frac{2e}{1+e})}$$ $$=\sqrt{\frac{2h}{g}(\frac{1-e}{1+e})}$$

## No fixed point Homeomorphism (TIFR 2013 problem 21)

Question:

True/False?

Every homeomorphism of the 2-sphere to itself has a fixed point.

Hint:

$$z= -z$$ implies $$z=0$$

Discussion:

2-sphere means $$S^2=\left \{(x,y,z)\in\mathbb{R}^3 | x^2+y^2+z^2=1 \right \}$$.

i.e, $$S^2=\left \{v\in\mathbb{R}^3 | ||v||=1 \right \}$$.

$$||.||$$ denotes the usual 2-norm (Euclidean norm).

Let us try $$f:S^2\to S^2$$ defined by $$f(v)=-v$$ for all $$v\in\mathbb{R}^3$$.

The only vector in $$\mathbb{R}^3$$ that is fixed by $$f$$ is 0, which doesn’t lie in $$S^2$$.

We hope $$f$$ turns out to be a homeomorphism.

$$||f(v)-f(w)||=||-v+w||=||v-w||$$. So f is in fact Lipshitz function, so continuous.

$$f(f(v)=v$$ for all $$v\in\mathbb{R}^3$$. Therefore, $$f$$ itself is inverse of $$f$$. Which proves that $$f$$ is bijective (since, inverse function exists) and homeomorphism (inverse is also continuous).